Circular Motion Calculator

Related Calculators

Uniform circular motion: Motion around a circle at constant speed, even though the direction keeps changing.[2]

Period, T: The time for one full revolution, measured in seconds.[1]

Frequency, f: The number of full revolutions completed each second, measured in hertz.[1]

Radius, r: The distance from the center of the circle to the moving object.

Angular speed, omega: How fast the angle changes as the object goes around the circle, usually in rad/s.

Linear speed, v: How fast the object moves along the circular path.

Centripetal acceleration: The inward acceleration needed to keep the object moving in a circle.[2]

Path circumference: The distance around the circle for one full lap.

Mistake: Entering values into hidden fields and expecting them to affect the result.
Fix: Only the visible required fields for your chosen "What do you know?" mode are used. Change the mode first, then fill in the matching visible inputs.

Mistake: Typing 0 or a negative number for "Radius, r (m)" or "Period, T (seconds)."
Fix: Enter a positive number greater than 0. Uniform circular motion here uses positive magnitudes only.

Mistake: Mixing up "Frequency, f (hertz)" and "Period, T (seconds)."
Fix: Use "Frequency, f (hertz)" for turns each second and "Period, T (seconds)" for seconds per turn. They are reciprocals, so a larger frequency means a smaller period.

Mistake: Entering "Angular speed, omega (rad/s)" in degrees per second while the input expects radians per second.
Fix: Convert your input to rad/s before entering it. The "Angular output unit" setting changes display only, not the required input unit.

Mistake: Expecting "Linear speed, v (m/s)" to be the same thing as "Centripetal acceleration."
Fix: "Linear speed" tells how fast the object moves along the circle, while "Centripetal acceleration" tells how strongly it turns inward to stay on that circle.

Mistake: Thinking a bigger "Path circumference" always means faster motion.
Fix: "Path circumference" depends on "Radius" only. To judge how fast the motion repeats, compare "Period," "Frequency," "Angular speed," and "Linear speed" together.

• This calculator assumes uniform circular motion, so the speed stays constant while only the direction changes.
• It works with magnitudes only. It does not assign clockwise or counterclockwise direction, and "Angular speed" is shown as a positive amount.
• Inputs must use the units shown in the labels: seconds, hertz, rad/s, m/s, and meters.
• The "Angular output unit" option changes display only. Internal calculations still use radians.
• Zero or negative values for the required visible inputs are invalid because they would make the motion undefined or nonphysical in this model.
• Results are rounded for display based on the selected format and significant figures, so shown values may differ slightly from full-precision math.
• This tool does not calculate forces or mass-related quantities. "Centripetal acceleration" is included, but force would need extra information such as mass.

What the calculator solves

The calculator uses the selected known pair to build one complete uniform circular motion state: radius, period, frequency, angular speed, linear speed, centripetal acceleration, and path circumference. Period and frequency describe one full revolution and are reciprocal ideas in uniform circular motion.[1]

Core formulas

f = 1 / T

ω = 2π / T

ω = 2πf

v = ωr

v = 2πr / T

r = vT / (2π)

r = v / (2πf)

a_c = v^2 / r

C = 2πr

ω in deg/s = ω in rad/s x 180 / π

How each mode works

If you choose period and radius, the calculator finds frequency from the period, then angular speed, then linear speed, then centripetal acceleration and circumference. If you choose frequency and radius, it first converts frequency to period and angular speed. If you choose angular speed and radius, it finds linear speed directly from v = ωr. If you choose linear speed with period or frequency, it solves radius first, then computes the remaining outputs. Hidden fields are ignored completely so they cannot change the answer by mistake.

Worked mini-example

Suppose you know "Linear speed, v (m/s)" = 10 and "Period, T (seconds)" = 5.

r = vT / (2π) = 10 x 5 / (2 x 3.141592653589793) = 7.9577 m

f = 1 / T = 1 / 5 = 0.2 Hz

ω = 2π / T = 1.2566 rad/s

a_c = v^2 / r = 100 / 7.9577 = 12.5664 m/s^2

C = 2πr = 50 m

That means the object travels a 50 m circle, completes 0.2 turns each second, and needs an inward acceleration of about 12.57 m/s^2 to stay on the path. In uniform circular motion, the speed can stay constant while the velocity direction keeps changing, which is why centripetal acceleration is still present.[2]

Assumptions used in the math

The formulas assume a perfect circle and constant speed. They also use positive magnitudes only, so the calculator does not track rotation direction. Angular output may be shown in degrees per second for readability, but the main calculations use radians internally.